Finite Difference Implementation

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Physical Audio Signal Processing

High-Accuracy Piano-String Modeling

Nonlinear Piano-String Equations of Motion in State-Space Form

Commuted Piano Synthesis

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According to Hooke's law, the force F exerted by a spring stretched a distance x from its rest position is F=-kx. — Click for http://en.wikipedia.org/wiki/Spring_%28device%29

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The sampling rate of a discrete-time signal is defined as the number of samples per second. Its units are thus in Hertz (Hz). Shannon's Sampling Theorem states that the original continuous-time signal can be recovered exactly from the samples if and only if the sampling rate is higher than twice the highest frequency present in the original signal. Any higher frequencies will alias to frequencies below half the sampling rate. — Click for https://ccrma.stanford.edu/~jos/mdft/Sampling_Theory.html

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Pressing a key on the piano causes a felt-tipped hammer to strike a vibrating string. — Click for http://www.speech.kth.se/music/5_lectures/

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Physical Audio Signal Processing

High-Accuracy Piano-String Modeling

Nonlinear Piano-String Equations of Motion in State-Space Form

Commuted Piano Synthesis

Finite Difference Implementation

Digitizing $\underline{{\ddot x}}_i(t)$ via the centered second-order difference [Eq.(7.5)]

$\displaystyle \underline{{\ddot x}}_n\isdefs \underline{x}_{n+1} - 2\underline{x}_n + \underline{x}_{n-1}$

we obtain the following explicit finite-difference scheme (§D.1):

$\displaystyle \underline{X}_{n+1} \eqsp \left[2\mathbf{I}+ \mathbf{M}^{-1}\mathbf{A}\right]\underline{X}_n - \underline{X}_{n-1} + B\underline{u}_n,$

where $\underline{u}_n$ is the external input vector for exciting the string (driving three adjacent masses in simulations to date).

Note that requiring three adjacent spatial string samples to be in contact with the piano hammer during the attack (which helps to suppress aliasing of spatial frequencies on the string during the attack) implies a sampling rate in the vicinity of 6 megahertz [267]. Thus, the model is expensive to compute! However, results to date show a high degree of accuracy, as desired. In particular, the stretching of the partial overtones in the stiff-string model of Fig. has been measured to be highly accurate despite using only three spring attachment points on one side of each mass disk [267].

See [53] for alternative finite-difference formulations that better preserve physical energy and have other nice properties worth considering.

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

Finite Difference Implementation

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4 Copyright © 2024-06-28 by Julius O. Smith III Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University